Sixth-order finite difference scheme for the Helmholtz equation with inhomogeneous Robin boundary condition

Zhang, Yang; Wang, Kun; Guo, Rui

doi:10.1186/s13662-019-2304-0

Cited by 8 publications

(7 citation statements)

References 31 publications

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“…Summarizing, the set of algebraic equations ( 22), ( 25)-( 26), ( 29)- (30), the discrete versions of the continuity of the scattered field (11) and the boundary condition at the obstacle (10) form the fourth order Karp DC system of linear equations to be solved. We denote this system as KDC4.…”

Section: Fourth Order DC Approximation At the Artificial Boundarymentioning

confidence: 99%

“…where F p−2 l−1, j and G p−2 l−1, j are part of the previously calculated (p − 2)th ordered numerical solution, and the discrete operators D p+2−q qθ are centered finite difference operators. Summarizing, the set of equations ( 32), ( 36)-( 37), ( 38)- (39), the discrete version of the continuity of the scattered field (11), and the appropriate discretization of the boundary condition at the obstacle (10) form the pth order DC discrete system of equations to be solved. We denote this system as KDCp.…”

Section: Arbitrary Order DC Approximation For the Kfementioning

confidence: 99%

“…The DC approximation of the governing Helmholtz equation of arbitrary order p is given by (33), while the pth order DC approximation of the Karp's farfield expansion ABC with enough NKFE terms is given by the discrete equations ( 36)-( 37), ( 38)- (39). The algebraic linear system for the scattered field obtained from all these discretizations is completed by the discrete equations corresponding to the continuity of the scattered field (11) at the artificial boundary and the appropriate discretization of the boundary condition at the obstacle (10). In the case case of Neumann or Robin condition, a pth order DC finite difference discretization of the boundary condition at the obstacle should be constructed.…”

Section: Concluding Remarks and Future Workmentioning

confidence: 99%

“…It resembles the strategy followed by Collatz and Leveque in [9,10], respectively, to obtain the well-known compact 9-point finite difference formula for the two-dimensional Poisson equation in cartesian coordinates. More recently, Zhang et al [11] derived a sixth order finite difference scheme for the Helmholtz equation with inhomogeneous Robin boundary conditions in two dimensions.…”

Section: Introductionmentioning

confidence: 99%

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High order methods for acoustic scattering: Coupling farfield expansions ABC with deferred-correction methods

et al. 2020

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Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs) with finite difference methods for the Helmholtz equation. These ABCs are based on exact representations of the outgoing waves by means of farfield expansions. The finite difference methods, which are constructed from a deferred-correction (DC) technique, approximate the Helmholtz equation and the ABCs, with the appropriate number of terms, to any desired order. As a result, high order numerical methods with an overall order of convergence equal to the order of the DC schemes are obtained. A detailed construction of these DC finite difference schemes is presented. Additionally, a rigorous proof of the consistency of the DC schemes with the Helmholtz equation and the ABCs in polar coordinates is also given. The results of several numerical experiments corroborate the high order convergence of the novel method.

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Section: Fourth Order DC Approximation At the Artificial Boundarymentioning

confidence: 99%

Section: Arbitrary Order DC Approximation For the Kfementioning

confidence: 99%

Section: Concluding Remarks and Future Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High order methods for acoustic scattering: Coupling farfield expansions ABC with deferred-correction methods

et al. 2020

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“…The number of points used in the proposed stencil varies from 9 in [6,35], 13 in [10], and both 17 and 25 in [9]. Other studies on FDMs that do not explicitly consider the numerical dispersion are [3] (a 4th order compact FDM on polar coodinates), [4] (a 4th order compact FDM), [32] (a 6th order compact FDM), and [36] (a 6th order FDM with non-compact stencils for corners and boundaries). The authors in [29] proposed a 3rd order compact immersed interface method for our model problem.…”

Section: Introduction and Motivationsmentioning

confidence: 99%

Sixth Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

Feng¹,

Han²,

Michelle³

2021

Preprint

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Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a sixth order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several stateof-the-art finite difference schemes in the literature, particularly in the critical pre-asymptotic region where kh is near 1 with k being the wavenumber and h the mesh size.

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A new development of sixth order accurate compact scheme for the Helmholtz equation

Kumar

Dubey

2019

J. Appl. Math. Comput.

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In this work, a new compact sixth order accurate finite difference scheme for the two and three-dimensional Helmholtz equation is presented. The main significance of the proposed scheme is that its sixth order leading truncation error term does not explicitly depend on the associated wave number. This makes the scheme robust to work for the Helmholtz equation even with large wave numbers. The convergence analysis of the new scheme is given. Numerical results for various benchmark test problems are given to support the theoretical estimates. These numerical results confirm the accuracy and robustness of the proposed scheme.

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Sixth-order finite difference scheme for the Helmholtz equation with inhomogeneous Robin boundary condition

Cited by 8 publications

References 31 publications

High order methods for acoustic scattering: Coupling farfield expansions ABC with deferred-correction methods

High order methods for acoustic scattering: Coupling farfield expansions ABC with deferred-correction methods

Sixth Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

A new development of sixth order accurate compact scheme for the Helmholtz equation

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